Can you make sense of these three proofs of Pythagoras' Theorem?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?

Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

My train left London between 6 a.m. and 7 a.m. and arrived in Paris between 9 a.m. and 10 a.m. At the start and end of the journey the hands on my watch were in exactly the same positions but the. . . .

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Think of a number and follow the machine's instructions... I know what your number is! Can you explain how I know?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.

Can you explain what is going on in these puzzling number tricks?

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

The heptathlon is an athletics competition consisting of 7 events. Can you make sense of the scoring system in order to advise a heptathlete on the best way to reach her target?

What is special about the difference between squares of numbers adjacent to multiples of three?

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

There are unexpected discoveries to be made about square numbers...

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

A task which depends on members of the group noticing the needs of others and responding.

How good are you at finding the formula for a number pattern ?

Surprising numerical patterns can be explained using algebra and diagrams...

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

Show that all pentagonal numbers are one third of a triangular number.

Can you find a rule which connects consecutive triangular numbers?

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

If you know the perimeter of a right angled triangle, what can you say about the area?